'Crystal Symmetry' printed from http://nrich.maths.org/

Initially you may find it hard to picture the lattice structure
from the given information. The diagram of the body centred cubic
lattice below should hopefully aid in your understanding: note how
it fulfils the criteria for the integer lattice.

Thus, the l axis is unchanged, but the m axis is rotated onto the n
axis, and the n axis onto the negative m axis. In essence this is a
rotation of 90$^\circ$ about the l axis in a right-handed sense. It
can be easily imagined that every Caesium ion maps straight to
another Caesium ion, and that every Chloride ion maps to another
chloride ion. Therefore, the crystal structure is preserved.

Having already done one matrix transformation, we now recognise
that the first column of the matrix is what the unit vector along
the l axis is tranformed onto, the second column for the m axis and
the third column for the n axis.

Therefore, this transformation represents a stretch of scale factor
3 along the l axis, and 2 along the n axis. This clearly does not
preserve the structure of the lattice as the ions are now no longer
evenly spaced along each axis!

From the previous two examples, we have reasoned that a matrix will
only preserve the structure of the lattice exactly if it is a pure
rotation matrix, with no stretching. A necessary condition for a
matrix to be a rotation (but not sufficient) is for columns of the
matrix to be orthonormal. Clearly it can be seen from the matrix
that the dot product of any of the columns is not zero, and so the
columns cannot be orthogonal. Thus, the matrix cannot be a pure
rotation, and so will not leave the lattice preserved
exactly.

The addition of this vector to the lattice moves each point by 1
unit on the l axis. Although this will not affect the spacing
between the atoms in the lattice, it will cause a misaligned, as
the first plane of caesium atoms will now be inline with where the
first plane of chloride ions were; the first plane of chloride ions
will not be in line with where the second plane of caesium ions
were etc.

This vector moves the caesium ion at the origin to the position (1,
1, 0) where there was not an ion previously. Therefore, although
the spacing in the lattice is preserved, its displacement is
noticeable, and so the structure is not completely preserved.

This vector corresponds to caesium ions being moved into the place
of another caesium ion (diagonally opposed), and chloride ions
doing the same. Thus, the structure is indistinguishable from
before, and so is preserved.

It can quickly be seen that this translation preserves the lattice
since the unit cube of the lattice is of side 2, and that the
vector is causing a translation along each of the axes by a
multiple of two. Thus caesium ions will always map onto caesium
ions, and chloride ions on chloride ions. Thus the lattice will be
preserved.

The types of transformation which leave the crystal invariant
are:
- Translations using vectors that have components which are
multiples of two. This way, caesium ions will always map onto
caesium ions and chloride ions onto chloride ions.
- Rotations using matrices. Rotation matrices must have orthonormal
columns and must have a determinant of 1.